The word token is used with several meanings.
From an Old English verb meaning “to show,” in the broadest sense a token is “something that serves to indicate a fact; a sign or a symbol.” The sign could be miraculous or merely evidence of something:
I do set my bow in the cloud, and it shall be for a token of a covenant between me and the earth. –Genesis, 9:13, KJV
God granted these holy men [the 100 fed miraculously by Elisha] a gracious token that the famine had lifted
The retiring employee received a watch as a token of the company’s appreciation.
As a physical object, a token may be a metal or plastic disk that serves to show that money has been paid for transportation or admission (e.g., a bus token). Board games like Monopoly include tokens used in play.
The expression “by the same token” means “for the same reason” or “in the same way.” Here are two correct examples of its use:
there was little evidence to substantiate the gossip and, by the same token, there was little to disprove it –example, OxfordDictionaries
because his mind is flexible it responds quickly … to what is before it, and by the same token it can call up from within a host of appropriate ideas example, Merriam-Webster
As is happening to many venerable expressions in this age of limited reading of traditional literature, “by the same token” is being altered by speakers who aren’t quite sure how to use it:
However at the same token, this same conversation could apply to couples who go through that process together…
In the same token, it cannot disregard basic issues of translation theory.
Does my body include the oxygen I am about to inhale and, on the same token, should I include the air I am about to exhale?
Not only is the preposition by being incorrectly replaced by at, in or on, the meaning is being lost:
I don’t dislike dogs (or other animals), but by the same token, I don’t want them in my house.
Here the meaning seems to be “on the other hand.”
With a documentation date of 1463, “by the same token” has had a good run in English. Could be that the expression–for some speakers at least–has reached retirement age.